Let $X_j$ be a D11: Set for each $j \in J$.
Then $X = \{ X_j \}_{j \in J}$ is an N-wise disjoint set collection with respect to $N \in 2, 3, 4, \ldots$ if and only if $$\begin{equation} \forall \, j_1, \ldots, j_N \in J \left( \forall \, n, m \in 1, \ldots, N \left( n \neq m \quad \implies \quad j_n \neq j_m \right) \quad \implies \quad \bigcap_{n = 1}^N X_{j_n} = \emptyset \right) \end{equation}$$